We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of $L^\infty$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space $C^{1-2α}$. Higher regularity starting from $C^δ$ with $δ>1-2α$ can be established through Besov space techniques and will be presented elsewhere \cite{CW6}.
We prove that every quasiconformal mapping from the harmonic $β$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $α$-Hölder continuous for $α<1-β$, with the Hölder coefficient that does not depend neither on the mapping nor on $β$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $α<1$. This extends some results from the complex plane obtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalaj in \cite{Kalaj6} for quasiconformal harmonic mappings.
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Ampère type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.
Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
Université Paris Cité, CNRS, Sorbonne Université（巴黎西岱大学、法国国家科学研究中心、索邦大学）；O. Mayer Mathematics Institute, Romanian Academy（罗马尼亚科学院 O. Mayer 数学研究所）；Univ Rennes, INSA Rennes, CNRS（雷恩大学、雷恩国立应用科学学院、法国国家科学研究中心）
This article addresses the regularity issue for minimizing fractional harmonic maps of order $s\in(0,1/2)$ from an interval into a smooth manifold. Hölder continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then Hölder continuity holds everywhere.
The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is uniformly elliptic, H$\ddot o$lder continuous and satisfies the natural structure conditions of fractional order, but without the concavity assumption of $F$. These assumptions are weaker and the result is stronger than that of Caffarelli and Wang[1], Chen[2].
We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation's criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. Combining the nonlinear maximum principle with a modulus of continuity, we prove a uniform-in-time gr…
We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations %$Δu-\frac{A}{|x|^{2+β}}u=0,\,\,(β\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01} \partial_{i} (a_{ij}(x) \partial_{j}u(x)) - \frac{A}{|x|^{2+β}} u(x) =0\quad (A>0,\quadβ\geq 0), \end{equation} \begin{equation}\label{02} \partial_{i} (a_{ij}(x,t) \partial_{j}u(x,t)) - \frac{A}{|x|^{2+β}} u(x,t)-\partial_{t}u(x,t) =0\quad (A>0,\quadβ\geq 0), \end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. …
In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type $ \sum_{i=1}^s \partial_{ii} u+ \sum_{i=s+1}^N \partial_i \bigg(A_i(x,u,\nabla u) \bigg) =0,$ for $x \in Ω\subset \subset \mathbb{R}^N$ and $1\leq s \leq N-1$, where each operator $A_i$ behaves directionally as the singular $p$-Laplacian, $1< p < 2$ and the supercritical condition $p+(N-s)(p-2)>0$ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we e…
We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we prove that the solution becomes immediately differentiable with Holder continuous derivatives. Therefore, the solutions to the equation are classical.
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high freq…
We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial regularity theorem for the Navier-Stokes equations. The key idea is to find a quantitative counterpart for the absolute continuity of the dissipation energy using the pigeonhole principle. Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending exponentially on the natural local energies of the solution. Then, we give two applications of the latter result in the axially symmetric case. The first one is a local quantitative…
We study the existence of weak solutions of (E) $ (-Δ)^αu+g(u)=ν$ in a bounded regular domain $Ω$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminusΩ$, where $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$, $ν$ is a Radon measure and $g$ is a nondecreasing function satisfying some extra hypothesis. When $g$ satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where $ν$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r|^{k-1}r$ with $k$ supercritical,…
We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. Such equations model the incompressible Navier-Stokes systems, generalized viscous Burgers-Hilbert equation and various active scalars. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for …
In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works …
